The wonderful ViaMichelin site says London to Bucharest is a driving distance of 2619km, with the majority of the journey (2114km) on motorways. The journey time quoted (using ViaMichelin's default settings for a hatchback car) is 27 hours 12 minutes.
Personally, travelling with an 8-year old, I wouldn't want to be travelling for more than 6 hours per day, so I'd suggest allowing a minimum of 5 days for the trip. If you're in a hurry then you could probably do it safely in 3 days but you'd almost certainly be exhausted by the time you got there. If time permits, then it would be better to spend a week on the trip because this would allow for time to relax and explore the countries which you're passing through.
I'm sorry that 'anywhere between 3 and 7 days' doesn't provide you with a precise answer but obviously I don't know what your priorities are. (In fact, I don't even know your starting point because you forgot to mention it!). All the same, I hope that this helps.
To plan your trip, go here:
http://www.viamichelin.com
Chris